However, the integral \(I+\frac{1}{2\pi i}\int _{\Sigma ^{mod}}\frac{(1-C_w)^{-1}Iw}{s-z}ds\)  is still hard to compute. Follow
the standard procedure of Deift-Zhou [1]}, we can separate the contributions from each saddle point. Before executing
this procedure, we need the following lemma.
